Abstract It is shown that every model of NF admits a permutation model containing an internal automorphism.

Abstract There are a variety of (“alternative”) axiomatic set theories available to mathematicians. It is worth asking how “alternative” they really are. Might they be no more than rephrasings of the theory (ZFC) that we already have? Here we give an account of the status of the Quine systems in this regard. Some are merely ZF in wolves’ clothing; some are genuine wolves.

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G. The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

For installing product line engineering in practice, strategies are needed that are composed of smaller steps where, on the one hand, each of them represents a well-defined move towards the ultimate goal of a well-functioning product line organization but, on the other hand, does not bear unmanageable risks. Hence product line approaches like Fraunhofer PuLSETM must define such steps, as well as provide a framework that supports their systematic combination into a tailored organizational strategy for installing product line engineering. This paper presents one step of such a strategy that focuses on the design activity. A well-known model-based design approach from the automotive domain is extended by the concept of variability and decision modeling. The resulting method has been prototyped and validated in a controlled environment. The measured results show that the method can be easily applied and leads to an effort distribution analogously to the typical product line curve.

Abstract A version of the Erdős-Rado theorem on partitions of the unordered n-tuples from uncountable sets is proved, without using the axiom of choice. The case with exponent 1 is just the Sierpinski-Hartogs' result that .

Abstract It is shown that, according to NF, many of the assertions of ordinal arithmetic involving theT-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix allurelemente.

Abstract Let ZFB be ZF + “every set is the same size as a wellfounded set”. Then the following are true. Every sentence true in every (Rieger-Bernays) permutation model of a model of ZF is a theorem of ZFB. (i.e., ZFB is the theory of Rieger-Bernays permutation models of models of ZF) ZF and ZFAFA are both extensions of ZFB conservative for stratified formulæ. The class of models of ZFB is closed under creation of Rieger-Bernays permutation models.

Abstract It is shown in ZF (without choice) that if there is a finite-to-one map (X) → X, then X is finite.

Abstract Tarski [5] showed that for any setX, its setω(X) of well-orderable subsets has cardinality strictly greater than that ofX, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation ∣ω(X)∣ = ∣Y∣. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation .

Abstract We consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a potentially useful new reduction of the consistency problem for Quine's NF, and a sharpening of another reduction of this problem due to Boffa.

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the form and these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.

The purpose of this paper is not to produce a survey of systems with a universal set: we do not yet understand them well enough. Rather it will concentrate on one particular aspect of them: the curious circumstance that there are half-a-dozen or so distinct proofs of ~ AC available in set theories with a universal set. This began to emerge in 1953 when Specker published in [2] a proof of ~ AC in Quine's system NF. Until recently this was an isolated phenomenon and poorly understood. The proof answered few questions and seemed rather ad hoc, thus inviting an investigation to determine whether this was an artefact caused by the particular axioms for NF, or part of a general conflict between the demands of big sets and AC. We will start with an informal discussion of the genesis of set theories with a universal set, collecting, en route, a number of desiderata for such theories. A number of new refutations of AC in systems meeting some or all of these conditions will then be presented. The conclusion the reader is invited to draw is that any sensible set theory with a universal set will probably have trouble with AC. Why do set theory with V e V at all? It contradicts conventional wisdom which teaches us that e is wellfounded. It must be said for this doctrine (let us abbreviate it to WOOF) that it has enabled the rapid execution of the logicist programme, so if that were the sole purpose of set theory we could consider ourselves well served. However most pure mathematicians are platonists and study things not because they are useful but simply because they are there. And the belief that V is not there can, after all, only be the result of early conditioning.

The permutation method was first applied to NF by Scott; other workers have published results ([2], [3]; or see [4] for a survey). Some of the results proved here have a more metalogical character than most previously yielded by this method. Hinnion and Petry (unpublished, but see [4]) proved that the existence of objects x such that x = {y: x∈y} is consistent with NF. (The significance of this is that the operation that sends x to {y: x∈y} respects ∈ and is thus an embedding.) It is demonstrated below that the existence of such objects is independent of the axioms of NF. The existence of nontrivial automorphisms of the universe is not an interesting possibility in ZF, since it contradicts wellfoundedness. Similar auguments are not available in NF, however, and it is shown below that if NF + AC for pairs is consistent, then we can consistently add an axiom stating that there is a ∈-automorphism of the universe that is a set of the model. (In the proof given below, the automorphism is in fact of order 2, but natural enrichments of the construction enable one to find automorphisms of other orders with suitable versions of choice as additional hypotheses.)